Optimal. Leaf size=25 \[ x \log \left (a c+\frac{b c}{x}\right )+\frac{b \log (a x+b)}{a} \]
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Rubi [A] time = 0.0115837, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2453, 2448, 263, 31} \[ x \log \left (a c+\frac{b c}{x}\right )+\frac{b \log (a x+b)}{a} \]
Antiderivative was successfully verified.
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Rule 2453
Rule 2448
Rule 263
Rule 31
Rubi steps
\begin{align*} \int \log \left (\frac{c (b+a x)}{x}\right ) \, dx &=\int \log \left (a c+\frac{b c}{x}\right ) \, dx\\ &=x \log \left (a c+\frac{b c}{x}\right )+(b c) \int \frac{1}{\left (a c+\frac{b c}{x}\right ) x} \, dx\\ &=x \log \left (a c+\frac{b c}{x}\right )+(b c) \int \frac{1}{b c+a c x} \, dx\\ &=x \log \left (a c+\frac{b c}{x}\right )+\frac{b \log (b+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0048467, size = 28, normalized size = 1.12 \[ \frac{(a x+b) \log \left (\frac{c (a x+b)}{x}\right )}{a}+\frac{b \log (x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.188, size = 44, normalized size = 1.8 \begin{align*} -{\frac{b}{a}\ln \left ({\frac{bc}{x}} \right ) }+x\ln \left ( ac+{\frac{bc}{x}} \right ) +{\frac{b}{a}\ln \left ( ac+{\frac{bc}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17715, size = 34, normalized size = 1.36 \begin{align*} x \log \left (\frac{{\left (a x + b\right )} c}{x}\right ) + \frac{b \log \left (a x + b\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93737, size = 63, normalized size = 2.52 \begin{align*} \frac{a x \log \left (\frac{a c x + b c}{x}\right ) + b \log \left (a x + b\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.341546, size = 20, normalized size = 0.8 \begin{align*} x \log{\left (\frac{c \left (a x + b\right )}{x} \right )} + \frac{b \log{\left (a x + b \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25673, size = 35, normalized size = 1.4 \begin{align*} x \log \left (\frac{{\left (a x + b\right )} c}{x}\right ) + \frac{b \log \left ({\left | a x + b \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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